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Chromatic aberrationsWhen different colors of light propagate at different speeds in a medium, the refractive index is wavelength dependent. This phenomenon is known as dispersion. A well- known example is the glass prism that disperses an incident beam of white light into a rainbow of colors [1]. Photographic lenses comprise various dispersive, dielectric glasses. These glasses do not refract all constituent colors of incident light at equal angles, and great efforts may be required to design an overall well-corrected lens that brings all colors together in the same focus. Chromatic aberrations are those departures from perfect imaging that are due to dispersion. Whereas the Seidel aberrations are monochromatic, i.e. they occur also with light of a single color, chromatic aberrations are only noticed with polychromatic light.Longitudinal and transverse chromatic aberrationOne discriminates between two types of chromatic aberration. Longitudinal chromatic aberration, also known as axial color, is the inability of a lens to focus different colors in the same focal plane. For a subject point on the optical axis the foci of the various colors are also on the optical axis, but displaced in the longitudinal direction (i.e. along the axis). This behavior is elucidated in Fig. 1 for a distant light source. In this sketch, only the green light is in sharp focus on the sensor. The blue and red light have a so-called circle of confusion in the sensor plane and are not sharply imaged.Figure 1. Origin of longitudinal chromatic aberration. The focal planes of the various colors do not coincide.Obliquely incident light leads to the transverse chromatic aberration, also known as lateral color. It refers to sideward displaced foci. In the absence of axial color, all colors are in focus in the same plane, but the image magnification depends on the wavelength. This behavior is illustrated in Fig. 2. The occurrence of lateral color implies that the focal length depends on the wavelength, whereas the occurrence of axial color in a complex lens does not strictly require a variable focal length. This seems counterintuitive, but in a lens corrected for longitudinal chromatic aberration the principal planes do not need to coincide for all colors. Since the focal length is determined by the distance from the rear principal plane to the image plane, the focal length may depend on the wavelength even when all images are in the same plane.Figure 2. Origin of transverse chromatic aberration. The size of the image varies from one color to the next.Figures 1 and 2 distinguish two simplified cases because in practice the longitudinal and lateral components are coexistent.A polychromatic subject fills a volume in the image space, which is comprised of a continuum of monochromatic images of various sizes and positions. Lateral color is particularly manifest in telephoto and reversed telephoto (retro focus) lenses. Chromatic aberrations often limit the performance of otherwise well-corrected telephoto designs. Lateral color, and not astigmatism, is the chief cause of the separation between the sagittal and tangential curves in their modulation transfer functions. The archetypal manifestation of chromatic aberrations is color fringing along boundaries that separate dark and bright parts of the image. This said, descriptions of the perceptible effects of chromatic aberrations do vary in literature. It is read that lateral color is a more serious aberration than axial color, because the former gives rise to colored fringes while the latter merely reduces the sharpness [2]. Oberkochen holds a different view and points to axial color as the most conspicuous color aberration [3]. Hecht describes the cumulative effect of chromatic aberrations as a whitish blur or hazed overlay [4]. The residual color errors of an optical system with achromatic (under)correction for axial color lead to a magenta halo or blur around each image point [5,6].ExamplesA cross was made from two black matchsticks and mounted on a cork. It was photographed with the Canon EF 85/1.2 L USM at a routine portrait distance of one meter. The cross was placed in the very image center, against a bright white background. The high contrast and the large working aperture of f/1.2 are responsible for the longitudinal chromatic aberration observed in Fig. 3. When auto focus is used to focus on the cross, there are purple fringes all around it. After slightly defocusing the lens manually, the color of the fringes becomes green. This green fringing is perhaps less distressing than the purple fringing, but it is still clearly visible. As the human eye and auto focus systems are particularly sensitive to green light, both manual focus and auto focus tend to bring the green image in sharp focus. The other colors of the spectrum are left defocused and add up to a magenta fringe. For this reason purple fringing is more common than green fringing.Figure 3. Examples of axial color (longitudinal chromatic aberration) with the Canon EF 85/1.2 L USM.A: auto focus on the cross. B: after slight manual defocus. The lens is used at full aperture on a Canon 5D.The same cross was photographed with three retro focus wide angle lenses (Fig. 4). It was placed close to the top-left image corner and tilted so as to point the long bar in the radial direction. This orientation places the crossbar in the tangential direction. For both Cosina lenses the crossbar evidences color fringes while the radial bar is unaffected. The joint presence of purple and green fringes at opposite sides of a tangential detail (lens A) are very characteristic of the transverse chromatic aberration. The bluish and yellowish fringes noticed with lens B are less common. Lens C is remarkably well corrected for a short-focus lens of the retro focus type. There is only the slightest hint of purple and green.Figure 4. Examples of lateral color (transverse chromatic aberration). All lenses are used at f/11 on a Canon 5D.A: Cosina 3.5-4.5/19-35 @ 20 mm. B: Cosina 3.8/20. C: Carl Zeiss Distagon 2.8/21. Longitudinal and lateral chromatic aberration can both give rise to colored edges, but properties are different. Axial color causes fringes all around objects, whereas lateral color only affects tangential details. Axial color can occur at any position in the image, whereas lateral color is absent in the image center and progressively worsens toward the image corners. Axial color is cured by stopping down the lens, whereas lateral color is present at all apertures. Chromatic aberration can manifest itself as a mixture of axial and later color, in which case it is partly cured by the use of a small aperture. Axial color yields fringes of a single color for an "in-focus" object (Fig. 3), whereas lateral color delivers two differently colored fringes at either side of the tangential structure (Fig. 4). Fringes due to axial color are particularly clear in parts of the scenery that are (just) out of focus, e.g. purple in front of the subject in sharp focus and green behind it, or vice versa. Although color fringing is only observed in places with a high contrast, both the longitudinal and transverse chromatic aberration degrade the overall resolution of a lens. Longitudinal chromatic aberration often operates in tandem with spherical aberration toshape the bokeh of a large-aperture lens. The combined effect of longitudinal chromatic aberration and spherical aberration is also known as spherochromatism [7,8].Secondary spectrumTransverse chromatic aberration leads to difficulties mostly in retro focus and telephoto lens designs, which are markedly asymmetrical, and longitudinal chromatic aberration in large-aperture lenses. An uncorrected design is called chromatic. The human eye is most sensitive to green light, and when a chromatic lens is focused for the green part of the visible spectrum, the blue and red ends are out of focus (see Fig. 5). The incorporation of simple achromatic doublets is quite effective against chromatic aberration. A famous example of such a doublet is the combination of a convex crown glass element with a concave flint glass element [9]. The achromatic correction of a photographic lens applies to two wavelengths toward the blue and red ends of the spectrum. Figure 5 shows a typical achromatic correction scheme. The zero crossings occur for the two matched wavelengths, the residual focus shift of the other colors is known as the secondary spectrum.Figure 5. Principles of color correction. The colored faces are known as the secondary spectrum.The advent of exotic glasses featuring low or anomalous dispersion enabled great progress in chromatic correction of photographic lenses. A design that brings three visible colors together is called apochromatic. A superachromatic lens corrects for four or more wavelengths and virtually eliminates color errors [10,11]. Strictly speaking it is not the number of zero crossings that determines the image quality, but the departures of the in-between wavelengths (the secondary spectrum). The designation APO is nowadays used by many a manufacturer to indicate such a reduced secondary spectrum, but true apochromatism is of rare occurrence in photographic lens designs. The typical correction curves shown in figure 5 are usually found with a 'focus shift' along the ordinate [2, 9,12] but sometimes with the 'focal length'. This may appear unimportant, but it makes a difference as the former case implies correction for axial color and the latter for lateral color. Likewise, the term secondary spectrum is most often encountered in relation to the longitudinal chromatic aberration. Nonetheless some authors use the term also to describe the transverse chromatic aberration [3,13] or explicitly plot correction curves with 'focal length' along the ordinate [7].Achromatic lensesThe achromatic design (Fig. 5) is the most common correction level in a photographic lens and accounts for the purple and green fringes observed in Fig. 3. When the green light is in proper focus on the sensor, red and blue are similarly defocused and add up to magenta. Conversely, when blue light is in proper focus the red light is also in focus, leaving only green defocused. Upon replacing "focus shift" by "focal length" along the ordinate in Fig. 5, the achromatic scheme also accounts for the fringes observed in Fig. 4 (case A) with purple toward the image center and green toward the image corner. For the case of transverse chromatic aberration a longer focal length corresponds to a larger image magnification. This implies that for the achromatic diagram in Fig 5, the blue and red images of a nonaxial detail are displaced slightly more outward from the image center than the green image. Figure 6 shows a cross, where the blue and red images are displaced upward relative to the green image. The addition of the three monochromatic images results in the aberrated cross at the right. Blue, green and red add up to white, blue and red add up to purple. To be sure, figure 6 is a simplified scheme of pure transverse chromatic aberration, with only three colors, and merely serves as an illustration. As it appears, the fringes are both inside and outside the original cross. The crossbar is broadened, but at the same time part of its interior is eaten away by the aberration.Figure 6. Origin of color fringing due to lateral color. The addition of the constituent colors leads to fringes in the right image.Purple fringing: lens or sensor?Chromatic aberration, and purple fringing in particular, have received considerable attention with the advent of digital cameras. Indeed, although chromatic aberrations can be noticed on film (Fig. 7), they do look more intense on CCD or CMOS images. On the one hand the digital photographer is only a few mouse clicks away from a full screen display, and few lenses stand image inspection at high magnification. On the other hand it is possible that digital imaging somehow renders the color fringes due to chromatic aberration more distinct. One of the proposed mechanisms is an enhanced spectral sensitivity in the (ultra)violet and (infra)red regimes, where lenses tend to be poorly corrected for chromatic aberration. Purple fringing is often blamed on sensor bloom, which is odd as blooming is a quite different phenomenon [14, 15]. In fact, there are as many arguments against sensor bloom as there are in favor of chromatic aberration to account for purple fringing. The examples shown on the present page are all demonstrably due to the lens and not to the sensor. Sensor bloom has no known color preference, and if it had, it would not change its colors upon defocusing a lens (as in Fig. 3) or upon swapping lenses (as in Fig. 4). The list of arguments is long [16].Figure 7. Chromatic aberrations recorded on film. The church cross is photographed off a slide projection screen; the vane, taken at the "best" focus and after slight defocusing, is a 2900 dpi scan obtained with a slide scanner. Notice the similarities with Figs. 3 and 4.© PA van Walree 2001-2007References[1] Isaac Newton, "A new theory about light and colours," Phil. Trans. 6, 3075-3087 (1672).[2] SPSE handbook of photographic science and engineering, edited by Woodlief Thomas Jr., John Whiley & sons.[3] Carl Zeiss, Camera Lens News 12 (2000).[4] Eugene Hecht, Optics, 3rd ed., Addison Wesley (1998).[5] Warren J. Smith, Modern optical engineering, 3rd ed., McGraw-Hill (2000).[6] Bruce H. Walker, "Understanding secondary color," Optical Spectra 12, 44-46 (1978).[7] Sidney F. Ray, Applied photographic optics, 3rd ed., Focal Press (2002).[8] Applied optics and optical engineering, Vol. III, edited by Rudolf Kingslake, Academic Press (1965).[9] Arthur Cox, Optics: the technique of definition, 6th ed., Focal Press (1946).[10] Max Herzberger and Nancy R. McClure, "The design of superachromatic lenses," Applied Optics 2, 553-560 (1963).[11] Hans Sauer, "Zeiss Sonnar F/5.6 250mm superachromat," The British Journal of Photography 123, 166-169 (1976).[12] Handbuch der Fototechnik, 9th ed., edited by Gerhard Teicher, VEB Fotokinoverlag Leipzig (1986).[13] Francis A. Jenkins and Harvey E. White, Fundamentals of optics, 4th ed., McGraw-Hill (1976).[14] /primer/digitalimaging/concepts/ccdsatandblooming.html[15] /ccd102.html[16] Ronald Parr, /~parr/photography/faq.html#purplefringespherical aberration | astigmatism and field curvature | distortion | chromatic aberrations | vignetting | lens hoods | flare | filter flare | depth of field | dof equations | vwdof | bokeh | spurious resolution | misconceptionsVignettingA photograph or drawing whose edges gradually fade into the surrounding paper is called a vignette. The art of creating such an illustration is a deliberate one. Yet the word vignetting is also used to indicate an unintended darkening of the image corners in a photographic image. There are three different mechanisms which may be responsible. Natural and optical vignetting are inherent to each lens design, while mechanical vignetting is due to the use of improper attachments to the lens. Natural and optical vignetting lead to a gradual transition from a brighter image center to darker corners. At large apertures both phenomena are present and the combined effect is often designated by the term 'illumination falloff'. Mechanical vignetting can also give rise to gradual falloff, although the usual connotation is one where it causes an abrupt transition with entirely black image corners.Optical vignettingMost photographic lenses exhibit optical vignetting to some degree. The effect is strongest when the lens is used wide open and will disappear when the lens is stopped down by a few stops. Together with natural vignetting, optical vignetting causes a gradual darkening of the image towards the corners. This illumination falloff often goes unnoticed but it may become disturbing when the subject has large faces with an even color or brightness. The steepness of the film also plays a role: the more contrasty the film, the more pronounced the effect.Figure 1 illustrates optical vignetting for the Planar 50/1.4 with an ever exciting subject like a brick wall. At full aperture the image reveals a 'hot spot': a brighter center and a darkening towards the corners (left photograph). When the lens is closed down to f/5.6, the light falloff has disappeared and an evenly illuminated wall remains (right photograph).Figure 1. Optical vignetting with the Planar 50/1.4. Left: f/1.4. Right: f/5.6.Optical vignetting is also known as artificial or physical vignetting. Its origin relates to the simple fact that a lens has a length. Obliquely incident light is confronted with a smaller lens opening than light approaching the lens head-on. In figure 2 the lens used to photograph the wall in figure 1 is shown at two f-stops and from two viewpoints. The white openings in the top illustrations denote the entrance pupil, which is the image of the aperture stop seen through all lens elements in front of it and from a position on the optical axis. The entrance pupil is the clear aperture for light that approaches the lens from the front and ends up in the image center.Figure 2. The Planar 50/1.4 at f/1.4 (left) and f/5.6 (right) seen from the optical axis (top) and from the semifield angle (bottom).The bottom illustrations show the lens from the semifield angle, which is half the full angle of view of the lens. Here, the white openings correspond to the clear aperture for light that is heading for the image corner. At f/1.4 the clear aperture is markedly reduced compared to the on-axis case: the entrance pupil is partially shielded by the lens barrel. More precisely, the aperture is delimited by the rims surrounding the front and rear elements. A smaller aperture implies that the lens collects less light for off-axis points than for on-axis points and hence that the image corners will be darker than the image center. At f/5.6 the entrance pupil is much smaller and no longer shielded by the lens barrel. Consequently, obliquely incident light sees the same aperture as normally incident light and there is no optical vignetting any more.Cat's eye effectThe consequences of optical vignetting for a subject that is in focus (cf. figure 1) is merely a reduced brightness towards the image corners. However, optical vignetting can also have a pronounced effect on out-of-focus parts of the image. Because the shape of an out-of-focus highlight (OOFH) mimics the shape of the clear aperture, the bottom left situation of figure 2 leads to the so-called cat's eye effect [1]. Figure 3 evidences the resemblance between the appearance of OOFH's and the aperture shape. With an increasing distance from the optical axis the shape of the OOFH progressively narrows and starts to resemble a cat's eye. The larger the distance from the image center, the narrower the cat's eye becomes.Figure 3. The cat's eye effect. The rectangular area indicated by the dotted white line is shown enlarged at the bottom. (Photograph by Peter Boehmer.)The cat's eye effect is readily observed in an SLR viewfinder. Just inspect distant street lights with the lens set atclose-focus. By judging the narrowness of the cat's eye with an OOFH in the image corner it is possible to estimate the amount of optical vignetting. If the lens is stopped down the aperture where optical vignetting disappears may also be visually found. For instance, the Planar 50/1.4 is almost completely cured at f/2.8.Optical vignetting tends to be stronger in wideangle lenses and large aperture lenses, but the effect can be noticed with most photographic lenses. Zoom lenses are often saddled with a fair amount of optical vignetting. Oversized front or rear elements help to reduce this type of vignetting and are frequently applied in wideangle lens designs. At any rate, the given speed of a lens always refers to the on-axis case; the full aperture for off-axis objects is smaller.Natural vignettingNatural vignetting, more properly termed natural light falloff, is inherent to each lens design and becomes more troublesome for wideangle lenses. It is associated with the famous cos4 law of illumination falloff. Contrary to popular belief, the argument for this law is not measured in object space but in image space. It is measured at the rear end of the lens as the angle at which the light impinges upon the film.To illustrate natural vignetting and to point out differences between lens designs at the same time, I consider two examples. While rangefinder wideangle lenses tend to be better corrected than their retrofocus competitors, in particular for curvilinear distortion and lateral color, they also have the disadvantage of persistent light fall-off. Two contemporary wideangle designs are shown in figure 4: the Carl Zeiss Distagon 21/2.8 and the Carl Zeiss Biogon 21/2.8. The top design is an asymmetrical retrofocus lens for Contax SLR cameras (and all that to give way to the mirror!), the bottom lens is a symmetrical design for use on the Contax G rangefinder cameras. The black bars indicate the actual position of the variable leaf diaphragm (the aperture stop), the red bars indicate the exit pupil, which is the image of the aperture stop that an observer sees when he looks into the lens from the rear. The exit pupil serves to illuminate the film and delineates the light cone received by a point on the film (figure 5).Figure 4. Image illumination in relation to the position of the exit pupil for two 21/2.8 lenses. Top: the retrofocus Distagon 21/2.8 design. Bottom: the Biogon 21/2.8 rangefinder design. The angle b differs between the designs.Figure 5. The cos4 law explained for the Biogon.Whereas the exit pupil of the rangefinder lens is separated from the film by approximately its focal length, the exit pupil of the retrofocus design is pushed away from the film with the glass. Thus, from the yellow cones illuminating the film, the angle b is significantly larger for the Biogon than for the Distagon. It is this angle that is most important for the cos4 law. An enlargement of the Biogon rear end is found in figure 5. Two cones are indicated, one to illuminate a point on the optical axis (image center) and one to illuminate an off-axis point (image corner). Compared with the image center, the corner illumination is less for three reasons. First, there is a cos2(b) factor due to the inverse square law: the light has a longer way to travel to the image corner. Second, the pupil seen by the off-axis point is not round but elliptical and has a smaller area than the round pupil seen by the image center. This yields another cos(b) factor. Note however that this cosine factor is approximate. It needs refinement when the pupil diameter is not small compared to its distance from the film [2]. Third, while the light hits the image center at normal incidence, it strikes the image corner at the angle b. This yields anothercos(b) factor. The last effect relates to Lambert's law and can be compared with a late afternoon sun which heats the earth less than the sun at noon because the same beam of sunlight is spread over a larger area. The combined effect of all cosine factors is a cos4 illumination falloff towards the image corners.It is remarked that the cos4 law is not truly a law but rather a combination of cosine factors which may or may not be all present in a given situation.Although the cos4 law is indeed better applied to image space than object space, for many (but not all) lenses good agreement between theory and practice is actually found by taking one cosine in object space and three cosines in image space. The single cosine in object space then accounts for the amount of light collected by the lens. If one presumes that all collected light is also projected on the film (conservation of energy), this one cosine replaces the second of the abovementioned factors. The resulting cos4 law then reads cos(a)*cos3(b), where a denotes the angle in object space. Significant departures from this law arise when lens designers use the Slyusarev effect [1] to increase the apparent size ofone or both pupils for off-axis points. This may result in a dramatic improvement of the image illumination.The angle a is the same for both 21-mm lenses, but since the angle b differs between the Distagon and the Biogon it is expected that the image illumination will differ between the two lenses. Indeed, there is a considerable difference: figure 6.Figure 6. Illumination charts for the Distagon 21/2.8 and the Biogon 21/2.8 (Zeiss data).In an illumination chart the curves are given relative to the image center. I.e, the image center illuminance is 1.0 and the illuminance at other points is given as a fraction relative to the image center. At full aperture both designs have a corner illuminance of 0.2, which means that the image center receives five times the amount of light the image corner receives. This will certainly be noticed in photographs which include a blue sky or the brick wall of figure 1. Upon stopping down the lenses, the Distagon image illumination improves drastically. At f/11 the corner is only 0.8 stops darker than the center. The Biogon illumination however does not improve that much, the corner illumination remaining 1.8 stops behind the center. A full stop difference remains between the corner illuminations of the two designs. The difference is that the Distagon suffers mostly from optical vignetting at full aperture and benefits from a smaller aperture. The Biogon on the other hand suffers primarily from natural vignetting, which is not cured by a smaller aperture. Figure 6 also shows the cos4 curves for both designs. There is an excellent match for the Distagon at small apertures. By contrast, the Biogon illuminance is close to its cos4 curve at all apertures.Mechanical vignettingWhen mechanical extensions to a lens protrude into its field of view, the image corners receive less light than they would in the absence of the extension and vignetting occurs. The extension can be too long a lens hood, stacked filters, or a combination. The hood and/or filters obscure the entrance pupil from obliquely incident light and the remedy is obvious: use proper accessories. A single, thick filter can already vignette a wideangle lens so it pays to check on vignetting behavior before filters and lens hoods are purchased.Figure 7. A typical case of mechanical vignetting. (Distagon 28/2 @ f/11 + Contax metal hood #3)Figure 7 illustrates a typical case of mechanical vignetting. The image was taken with the Distagon 28/2 equipped with Contax metal hood #3, which is simply too long for this lens. A graphical explanation is given in Figure 8. With the Distagon used at f/11 and infinity, an image corner relies for its illumination on the orange ray pencil, which comes from infinity heading towards the entrance pupil (in red). The angle with the optical axis is the semifield angle, which amounts to 37 degrees. In the absence of a hood the oblique ray pencil has full access to the entrance pupil, but in the presence of the hood the entrance pupil is invisible to this pencil. The pupil is eclipsed by the hood and the image corner receives no light at all. A comprehensive discussion on mechanical vignetting is found on the lens hood page.Figure 8. The Distagon 28/2 without and with Contax metal hood #3. The oblique ray pencil is blocked when the hood is attached.Note that figure 8 does not show refraction of the pencils. For mechanical vignetting it suffices to consider the entrance pupil and obstructions in front of the lens. Refraction is indirectly taken into account by the position and size of the entrance pupil (Zeiss data). However, the pencils are drawn correctly outside the lens.RemediesThree causes of dark image corners have passed in review. Optical vignetting is due to the dimensions of the lens: off-axis object points are confronted with a smaller aperture than a point on the optical axis. The remedy is to stop down the lens. The darkening noticed at full aperture already improves greatly when the lens opening is decreased by one stop. A complete cure often requires two or three stops - depending on the design. The darkening that remains at small apertures is due to natural vignetting. Natural vignetting is not cured by stopping down the lens. Lenses which strongly suffer from natural vignetting, such as the Carl Zeiss Hologon 16/8, benefit from a gradual gray filter which is dark in the center and brighter towards the corner. Finally, mechanical vignetting is due to extensions attached to a lens. The image corners may become completely black and the photographer can only blame himself as he should use proper accessories.All types of vignetting are at their worst with the lens focused at infinity. At close focus the field of view decreases and the image circle increases. The vignetted area is pushed outwards with the image circle and when the focus is close enough the optical or mechanical vignetting will be outside the film frame (or digital sensor). As a matter of fact, it is optical vignetting that determines the size of the image circle in the first place. Natural vignetting improves too, since the exit pupil moves away from the film (at least, with most lens designs).It should be mentioned that vignetting is not always a bad thing. A lens designer can deliberately introduce vignetting in favor of a better control of aberrations, sacrificing field coverage for overall contrast and sharpness. Also, the vignetting effect can be used artistically to draw the attention away from the image periphery and to emphasize the subject. Commercially available attachments like the Lindahl Low Key en High Key Vignetters are popular among portrait photographers.© PA van Walree 2002-2007References[1] Sidney F. Ray, Applied photographic optics, 2nd ed., Focal Press (1997).[2] Warren J. Smith, Modern optical engineering, 3rd ed., McGraw-Hill (2000).People who arrived on this page through wikipedia, or otherwise wonder about the unorthodox terminology ... the terms "natural vignetting", "optical vignetting", "artificial vignetting", "physical vignetting", "mechanical vignetting" and "cat's eye effect" have all been adopted from reference [1].spherical aberration | astigmatism and field curvature | distortion | chromatic aberrations | vignetting | lens hoods | flare | filter flare | depth of field | dof equations | vwdof | bokeh | spurious resolution | misconceptions。